Question

limit

Answer 1

if \[\lim_{x \rightarrow a} f(x)\] exists but \[\lim_{x \rightarrow a} g(x)\] does not exist ,show that both lim\[\lim_{x \rightarrow a} ( f(x)+g(x))\] and\[\lim_{x \rightarrow a} (f(x)-g(x))\] do not exist

Answer 2

use property of lim(a+b=lima+limb

Answer 3

so??

Answer 4

let take lim f(x) =a
then:
lim(f(x)+g(x))=lim f(x) + lim g(x)=a+ lim g(x) , doesn't exist , since lim g(x) does not.
Same for the other

Answer 5

same goes to minus???

Answer 6

yes

Answer 7

if \[\lim_{x \rightarrow a}\frac{ f(x) }{ x-a }\] exist ,prove that \[\lim_{x \rightarrow a}f(x) =0\]

Answer 8

You can't use those rules lim (f+g) = lim f + lim g because its only works when all the limits are defined. I think you need to argue by contradiction :

Suppose lim f exists and lim (f+g) exists. Then
lim (f+g) - lim(f) exists and = lim (f+g)-f = lim g so lim g exists also but we know it does not exist. hence limit of f+g can not exist.

Answer 9

lim (f/(x-a) = b and lim (x-a) = 0
therefore
lim (f) = lim (f/(x-a)) * lim (x-a) = b*0 = 0